128 ideas
| 2196 | The observation of human blindness and weakness is the result of all philosophy [Hume] |
| 2187 | If we suspect that a philosophical term is meaningless, we should ask what impression it derives from [Hume] |
| 2200 | All experimental conclusions assume that the future will be like the past [Hume] |
| 4636 | All reasoning concerning matters of fact is based on analogy (with similar results of similar causes) [Hume] |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| 2197 | Reason assists experience in discovering laws, and in measuring their application [Hume] |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| 7700 | We can't think about the abstract idea of triangles, but only of particular triangles [Hume] |
| 13602 | We cannot form an idea of a 'power', and the word is without meaning [Hume] |
| 2216 | We transfer the frequency of past observations to our future predictions [Hume] |
| 2215 | There is no such thing as chance [Hume] |
| 2209 | Belief is stronger, clearer and steadier than imagination [Hume] |
| 2208 | Belief is just a particular feeling attached to ideas of objects [Hume] |
| 2207 | Belief can't be a concept plus an idea, or we could add the idea to fictions [Hume] |
| 3661 | 'Natural beliefs' are unavoidable, whatever our judgements [Hume, by Strawson,G] |
| 2213 | Beliefs are built up by resemblance, contiguity and causation [Hume] |
| 2191 | Relations of ideas are known by thought, independently from the world [Hume] |
| 2239 | If secondary qualities (e.g. hardness) are in the mind, so are primary qualities like extension [Hume] |
| 2237 | It never occurs to people that they only experience representations, not the real objects [Hume] |
| 23631 | Hume is loose when he says perceptions of different strength are different species [Reid on Hume] |
| 2192 | All reasoning about facts is causal; nothing else goes beyond memory and senses [Hume] |
| 2246 | If books don't relate ideas or explain facts, commit them to the flames [Hume] |
| 2184 | All ideas are copies of impressions [Hume] |
| 2190 | All objects of enquiry are Relations of Ideas, or Matters of Fact [Hume] |
| 2182 | Impressions are our livelier perceptions, Ideas the less lively ones [Hume] |
| 2189 | All ideas are connected by Resemblance, Contiguity in time or place, and Cause and Effect [Hume] |
| 2702 | Only madmen dispute the authority of experience [Hume] |
| 2183 | We can only invent a golden mountain by combining experiences [Hume] |
| 2194 | How could Adam predict he would drown in water or burn in fire? [Hume] |
| 2205 | You couldn't reason at all if you lacked experience [Hume] |
| 2217 | When definitions are pushed to the limit, only experience can make them precise [Hume] |
| 2186 | We cannot form the idea of something we haven't experienced [Hume] |
| 3902 | Hume mistakenly lumps sensations and perceptions together as 'impressions' [Scruton on Hume] |
| 23421 | If a person had a gap in their experience of blue shades, they could imaginatively fill it in [Hume] |
| 2206 | Reasons for belief must eventually terminate in experience, or they are without foundation [Hume] |
| 2235 | There is no certain supreme principle, or infallible rule of inference [Hume] |
| 10328 | We think testimony matches reality because of experience, not some a priori connection [Hume] |
| 2230 | Good testimony needs education, integrity, motive and agreement [Hume, by PG] |
| 2238 | Reason can never show that experiences are connected to external objects [Hume] |
| 2242 | Mitigated scepticism draws attention to the limitations of human reason, and encourages modesty [Hume] |
| 2243 | Mitigated scepticism sensibly confines our enquiries to the narrow capacity of human understanding [Hume] |
| 2236 | Examples of illusion only show that sense experience needs correction by reason [Hume] |
| 2240 | It is a very extravagant aim of the sceptics to destroy reason and argument by means of reason and argument [Hume] |
| 2241 | The main objection to scepticism is that no good can come of it [Hume] |
| 2198 | We assume similar secret powers behind similar experiences, such as the nourishment of bread [Hume] |
| 2202 | Fools, children and animals all learn from experience [Hume] |
| 2204 | All inferences from experience are effects of custom, not reasoning [Hume] |
| 2199 | Reason cannot show why reliable past experience should extend to future times and remote places [Hume] |
| 2201 | Induction can't prove that the future will be like the past, since induction assumes this [Hume] |
| 2203 | If we infer causes from repetition, this explains why we infer from a thousand objects what we couldn't infer from one [Hume] |
| 6350 | Premises can support an argument without entailing it [Pollock/Cruz on Hume] |
| 3598 | Hume just shows induction isn't deduction [Williams,M on Hume] |
| 2210 | A picture of a friend strengthens our idea of him, by resemblance [Hume] |
| 17712 | General ideas are the connection by resemblance to some particular [Hume] |
| 8544 | Hume does not distinguish real resemblances among degrees of resemblance [Shoemaker on Hume] |
| 2211 | When I am close to (contiguous with) home, I feel its presence more nearly [Hume] |
| 2214 | Our awareness of patterns of causation is too important to be left to slow and uncertain reasoning [Hume] |
| 2212 | An object made by a saint is the best way to produce thoughts of him [Hume] |
| 2222 | The doctrine of free will arises from a false sensation we have of freedom in many actions [Hume] |
| 2223 | Liberty is merely acting according to the will, which anyone can do if they are not in chains [Hume] |
| 3655 | Hume makes determinism less rigid by removing the necessity from causation [Trusted on Hume] |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| 2220 | Only experience teaches us about our wills [Hume] |
| 2224 | Praise and blame can only be given if an action proceeds from a person's character and disposition [Hume] |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| 2225 | If you deny all necessity and causation, then our character is not responsible for our crime [Hume] |
| 2226 | Repentance gets rid of guilt, which shows that responsibility arose from the criminal principles in the mind [Hume] |
| 2233 | No government has ever suffered by being too tolerant of philosophy [Hume] |
| 2195 | We can discover some laws of nature, but never its ultimate principles and causes [Hume] |
| 2245 | A priori it looks as if a cause could have absolutely any effect [Hume] |
| 4772 | If a singular effect is studied, its cause can only be inferred from the types of events involved [Hume] |
| 8341 | Hume never even suggests that there is no such thing as causation [Hume, by Strawson,G] |
| 8344 | At first Hume said qualities are the causal entities, but later he said events [Hume, by Davidson] |
| 2234 | It is only when two species of thing are constantly conjoined that we can infer one from the other [Hume] |
| 3662 | Hume says we can only know constant conjunctions, not that that's what causation IS [Hume, by Strawson,G] |
| 4771 | In both of Hume's definitions, causation is extrinsic to the sequence of events [Psillos on Hume] |
| 5194 | Hume's definition of cause as constantly joined thoughts can't cover undiscovered laws [Ayer on Hume] |
| 2221 | A cause is either similar events following one another, or an experience always suggesting a second experience [Hume] |
| 2193 | No causes can be known a priori, but only from experience of constant conjunctions [Hume] |
| 8422 | Cause is where if the first object had not been, the second had not existed [Hume] |
| 2218 | In observing causes we can never observe any necessary connections or binding qualities [Hume] |
| 15249 | Hume never shows how a strong habit could generate the concept of necessity [Harré/Madden on Hume] |
| 8339 | Hume's regularity theory of causation is epistemological; he believed in some sort of natural necessity [Hume, by Strawson,G] |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| 2244 | It can never be a logical contradiction to assert the non-existence of something thought to exist [Hume] |
| 2232 | You can't infer the cause to be any greater than its effect [Hume] |
| 2229 | To establish a miracle the falseness of the evidence must be a greater miracle than the claimed miraculous event [Hume] |
| 2227 | A miracle violates laws which have been established by continuous unchanging experience, so should be ignored [Hume] |
| 2228 | All experience must be against a supposed miracle, or it wouldn't be called 'a miracle' [Hume] |
| 2185 | The idea of an infinite, intelligent, wise and good God arises from augmenting the best qualities of our own minds [Hume] |